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Equality in Approximate Tolerance Geometry

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Book cover Intelligent Systems'2014

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 322))

Abstract

The framework of Approximate Tolerance Geometry (ATG) has been proposed in [1] as an approach to handling large and heterogeneous imperfections in geometric data in vector-based geographic information systems. Here, different types of positional error can often only be subsumed as possibilistic location constraints. The application of the ATG framework to a classical geometry provides a calculus for the propagation of this error type in geometric reasoning. As a first step towards an implementation of an ATG geometry, the paper applies the framework to the geometric equality relation. It thereby lays the basis for the application of ATG to the other axioms of classical geometry.

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References

  1. Wilke, G.: Fuzzy logic with evaluated syntax for sound geometric reasoning with geographic information. In: Proceedings of the Annual Meeting of the North American Fuzzy Information Processing Society, NAFIPS 2014 (2014)

    Google Scholar 

  2. Goodchild, M.F.: Citizens as voluntary sensors: Spatial data infrastructure in the world of web 2.0. International Journal of Spatial Data Infrastructures Research 2, 24–32 (2007)

    Google Scholar 

  3. Wilke, G.: Towards approximate tolerance geometry for gis - a framework for formalizing sound geometric reasoning under positional tolerance. Ph.D. dissertation, Vienna University of Technology (2012)

    Google Scholar 

  4. Pullar, D.V.: Consequences of using a tolerance paradigm in spatial overlay. In: Proceedings of the Auto-Carto 11, pp. 288–296 (1993)

    Google Scholar 

  5. Novak, V., Perfilieva, I., Mockor, J.: Mathematical Principles of Fuzzy Logic. Kluwer Academic Publishers (1999)

    Google Scholar 

  6. Gerla, G.: Fuzzy Logic - Mathematical Tools for Approximate Reasoning. In: Trends in Logic, vol. 11. Kluwer Acadeic Publishers (2001)

    Google Scholar 

  7. Gerla, G.: Approximate similarities and poincare paradox. Notre Dame Journal of Formal Logic 49(2), 203–226 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Perkal, J.: On epsilon length. Bulletin de l’Academie Polonaise des Sciences 4, 399–403 (1956)

    MATH  MathSciNet  Google Scholar 

  9. Shi, W.: A generic statistical approach for modelling error of geometric features in gis. International Journal of Geographical Information Science 12(2), 131–143 (1998)

    Article  Google Scholar 

  10. Buyong, T.B., Frank, A.U., Kuhn, W.: A conceptual model of measurement-based multipurpose cadastral systems. Journal of the Urban and Regional Information Systems Association URISA 3(2), 35–49 (1991)

    Google Scholar 

  11. Shi, W., Cheung, C.K., Zhu, C.: Modelling error propagation in vector-based buffer analysis. International Journal of Geographical Information Science 17(3), 251–271 (2003)

    Article  Google Scholar 

  12. Leung, Y., Ma, J.-H., Goodchild, M.F.: A general framework for error analysis in measurement-based gis, parts 1-4. Journal of Geographical Systems 6(4), 323–428 (2004)

    Article  Google Scholar 

  13. Dougenik, J.A.: Whirlpool: A geometric processor for polygon coverage data. In: Proceddings AUTOCARTO, vol. 4, pp. 304–311 (1979)

    Google Scholar 

  14. Beard, K.M.: Zipping: New software for merging map sheets. In: Proceedings of the American Congress on Surveying and Mapping, vol. 1, pp. 153–161 (1986)

    Google Scholar 

  15. Chrisman, N.R.: The accuracy of map overlays: A reassessment. In: Landscape and Urban Planning, vol. 14, pp. 427–439 (1987), http://www.sciencedirect.com/science/article/B6V91-46MN01M-1X/2/297d236facc206cdf028eedd653eb36d

  16. Beard, K.M., Chrisman, N.R.: Zipper: A localized approach to edgematching. Cartography and Geographic Information Science 15(2), 163–172 (1988)

    Article  Google Scholar 

  17. Chrisman, N., Dougenik, J.A., White, D.: Lessons for the design of polygon overlay processing from the odyssey whirlpool algorithm. In: Bresnahan, P., Corwin, E., Cowen, D. (eds.) Proceedings of the 5th International Symposium on Spatial Data Handling, vol. 2, pp. 401–410. IGU Commission of GIS (1992)

    Google Scholar 

  18. Zhang, G., Tulip, J.: An algorithm for the avoidance of sliver polygons and clusters of points in spatial overlay. In: Proceedings of the 4th Inthernational Symposium on Spatial Data Handling (1990)

    Google Scholar 

  19. Harvey, F., Vauglin, F.: Geometric match processing: Applying multiple tolerances. In: Proceedings of 7th International Symposium on Spatial Data Handling (1996)

    Google Scholar 

  20. Abdelmoty, A.I., Jones, C.B.: Towards maintaining consistency of spatial databases. In: Proceedings of the Sixth International Conference on Information and Knowledge Management, CIKM 1997, pp. 293–300. ACM, New York (1997), http://doi.acm.org/10.1145/266714.266913

    Chapter  Google Scholar 

  21. Clementini, E.: A model for uncertain lines. Journal of Visual Languages & Computing 16(4), 271–288 (2005)

    Article  Google Scholar 

  22. Salesin, D., Stolfi, J., Guibas, L.: Epsilon geometry: building robust algorithms from imprecise computations. In: SCG 1989: Proceedings of the Fifth Annual Symposium on Computational Geometry, pp. 208–217. ACM, New York (1989)

    Google Scholar 

  23. Veregin, H.: Data quality parameters. In: Longley, P.A., Goodchild, M.F., Maguire, D.J., Rhind, D.W. (eds.) Geographical Information Systems, 2nd edn., vol. 1&2, pp. 177–189. John Wiley & Sons (1999)

    Google Scholar 

  24. Roberts, F.S.: Tolerance geometry. NDJFAM 14(1), 68–76 (1973)

    MATH  Google Scholar 

  25. Katz, M.: Inexact geometry. Notre Dame Journal of Formal Logic 21(3), 521–535 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hájek, P.: Fuzzy logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Fall 2010 edn. (2010), http://plato.stanford.edu/archives/fall2010/entries/logic-fuzzy/

  27. Godo, L., Rodriguez, R.O.: Logical approaches to fuzzy similarity-based reasoning: an overview, vol. 504, pp. 75–128. Springer (2008)

    Google Scholar 

  28. Wilke, G., Frank, A.: On equality of lines with positional uncertainty. In: Proceedings of the Sixth International Conference on Geomgraphic Information Science (2010)

    Google Scholar 

  29. Wilke, G., Frank, A.: Tolerance geometry - euclids first postulate for points and lines with extension. In: Proceedings of the ACM SIGSPATIAL 2010, San Jose, California, USA (2010)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Information Science, University of Applied Sciences and Arts Northwestern Switzerland, Riggenbachstr. 16, 4600, Olten, Switzerland

    Gwendolin Wilke

Authors
  1. Gwendolin Wilke
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Correspondence to Gwendolin Wilke .

Editor information

Editors and Affiliations

  1. School of Computing and Communications, Lancaster University, Lancaster, United Kingdom

    P. Angelov

  2. Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria

    K.T. Atanassov

  3. Intelligent Systems Department, Bulgarian Academy of Sciences Inst. of Infor. & Communication Techn., Sofia, Bulgaria

    L. Doukovska

  4. Metalurgy, University of Chemical Technology and, Sofia, Bulgaria

    M. Hadjiski

  5. University of Library Studies and IT (ULSIT), Sofia, Bulgaria

    V. Jotsov

  6. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    J. Kacprzyk

  7. Knowledge Engineering and Discovery Research Institute, Auckland University of Technology, Auckland, New Zealand

    N. Kasabov

  8. Intelligent Systems Laboratory, Prof. Assen Zlatarov University Faculty of Technical Sciences, Bourgas, Bulgaria

    S. Sotirov

  9. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    E. Szmidt

  10. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    S. Zadrożny

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© 2015 Springer International Publishing Switzerland

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Wilke, G. (2015). Equality in Approximate Tolerance Geometry. In: Angelov, P., et al. Intelligent Systems'2014. Advances in Intelligent Systems and Computing, vol 322. Springer, Cham. https://doi.org/10.1007/978-3-319-11313-5_33

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  • DOI: https://doi.org/10.1007/978-3-319-11313-5_33

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-11312-8

  • Online ISBN: 978-3-319-11313-5

  • eBook Packages: EngineeringEngineering (R0)

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